CNRS researcher at the university of Paris 13.

I am interested in algebraic topology, algebraic geometry, and anything in between.

CNRS researcher at the university of Paris 13.

I am interested in algebraic topology, algebraic geometry, and anything in between.

and two Ph.D students:

Some of the work in progress includes:

#### The real cyclotomic trace map

## With Jay Shah and Thomas Nikolaus,

*in preparation*.## We identify the geometric fixed points of real topological cyclic homology in terms of normal L-theory, and deduce a real version of the Dundas-Goodwillie-McCarthy theorem in algebraic K-theory. This is part of work building towards the constructing of efficient trace methods for the study of hermitian K-theory.

#### The motivic hermitian K-theory spectrum

## With Baptiste Calmès and Denis Nardin,

*in preparation*.## We construct motivic spectra representing (homotopy invariant) Grothendieck-Witt and L-theory over general base schemes. When the base scheme is regular Noetherian of finite Krull dimension these motivic spectra represent symmetric Grothendieck-Witt and L-theory.

#### Hermitian trace formulas for singularity categories

## With Tasos Moulinos and Ran Azouri

## This project involves generalizing work of Toën and Vezzosi on the conductor formula and the trace of singularity categories for a degeneration of a family of smooth and proper schemes, where now the singularity category is endowed with a Poincaré structure, and the conductor formula is expressed in terms of the motivic trace of the generic and special fibres, which take values in their respective Grothendieck-Witt groups.

#### Verdier duality and assembly maps

## With Guglielmo Nocera

## In this project we explore a Poincaré categorical approach to the study of the assembly map in L-theory and Grothendieck-Witt theory, pursuing the ideas apprearing in the fourth installement of the nine author project concerning the assembly map of the circle.

#### Topological Hochschild homology for stable and Poincaré categories

#### Chromatic behaviour of hermitian K-theory

- Appendix to
#### Stable Moduli spaces of hermitian forms (F. Hebestreit, W. Steimle)

## The Grothendieck-Witt space of a Poincaré ∞-category with a suitable weight structure is descibed in terms of a certain group completion involving only Poincaré objects in the heart. This results is what enables one to express classical Grothendieck-Witt theory of rings in terms of that of a suitable Poincaré ∞-category. In the appendix an alternative root to the main application is proposed via a weight theorem on the level of L-theory.

#### Hermitian K-theory of stable ∞-categories IV: Poincaré motives

## This is the fourth instalment of a four-paper nine-author project on hermitian K-theory. It contains the theory of Poincaré motives, localising invariants, and applications to multiplicative properties of Grothendieck-Witt and L-theory. We also revisit and generalize the Shansen splitting phenomenon and use it to show that the universal localising replacement of L-theory coincides with L-theory with universal decoration.

## With Baptiste Calmès, Emanuele Dotto, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle,

*in preparation*.#### Hermitian K-theory of stable ∞-categories III: Grothendieck-Witt groups of rings

## In this third instalment we prove the main results concerning Grothendieck-Witt and L-theory of rings. In particular, we solve the homotopy limit problem for number rings, calculate the various flavours of Grothendieck-Witt groups of the integers, prove finite generation results for Grothendieck-Witt groups of number rings and show that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of finite global dimension is an equivalence in suffiently high degrees.

## With Baptiste Calmès, Emanuele Dotto, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle,

*submitted*(pdf, arXiv).#### Hermitian K-theory of stable ∞-categories II: cobordism categories and additivity

## In this second instalment we develop a cobordism inspired approach to hermitian K-theory and use it to prove most of the core results of this project, including additivity and universality of the Grothendieck-Witt spectrum and its relation to L-theory via the Tate square. As a direct corollary we obtain an extension of Karoubi's fundamental theorem to general rings, affirming a conjecture of Karoubi and Giffen.

## With Baptiste Calmès, Emanuele Dotto, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle,

*submitted*(pdf, arXiv ).#### Hermitian K-theory of stable ∞-categories I: Foundations

## This is the first instalment of a four-part nine-author paper on hermitian K-theory in the setting of stable ∞-categories equipped with a

*Poincaré structure*. We develop the procedure of deriving quadratic functors which allows to produce the classical setting of forms over rings. We also develop in the detail the general case of ring spectra and construct Poincaré structures on parameterized spectra used to recover visible L-theory. We also make a thorough inverstigation of global structural properties of Poincaré ∞-categories, laying the necessary foundations for the subsequent instalments.## With Baptiste Calmès, Emanuele Dotto, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle,

*Selecta Mathematica*, 29.1, 2023, p. 1-269 (pdf, arXiv ).

#### Rational points on elliptic surfaces and squares represented by products of quadratic forms

## We use Swinnerton-Dyer's method to give sufficient conditions for the existence of rational points on K3 surfaces which are 2-coverings of the projective plain ramified over a union of three diagonal conics.

*In preparation*.#### Supersolvable descent for rational points

## We develop a technique for deducing results about rational points on quotients of varieties by actions of finite supersolvable groups by reducing the problem to the fibration method over tori.

## With Olivier Wittenberg,

*Submitted*.#### Rational points on fibrations with few non-split fibres

## We revisit the foundations of the fibration method and extract some new unconditional cases when the number of non-split fibres is at most 3.

## With Dasheng Wei and Olivier Wittenberg,

*Journal für die reine und angewandte Mathematik*, 2022.791, 2022, p. 89-133 (pdf, arXiv).#### The Massey vanishing conjecture for number fields

## We prove the n-fold Massey vanishing conjecture for all number fields and all n's.

## with Olivier Wittenberg,

*Duke mathematical Journal*, Advance Publication, 1-41, 2022 (pdf, arXiv).- Appendix to
#### Number fields with prescribed norms (C. Frei, D. Loughran and R. Newton)

## with Olivier Wittenberg,

*Commentarii Mathematici Helvetici*, 97.1,2022,p.133-181 (arXiv). #### Zéro-cycles sur les espaces homogènes et problème de Galois inverse

## with Olivier Wittenberg,

*Journal of the American Mathematical Society*, 33 (3), 2020, p. 775–805 (pdf, arXiv).#### Second descent and rational points on Kummer varieties

*Proceedings of the London Mathematical Society*, 118 (3), 2019, p. 606–648. (pdf, arXiv).#### Hasse principle for Kummer varieties

## with Alexei Skorobogatov,

*Algebra and Number Theory*, 10.4, 2016, p. 813-841 (pdf, arXiv).#### Geometry and arithmetic of certain log K3 surface

*Annales de l'Institut Fourier*, 67 (5), 2017, p. 2167-2200 (pdf, arXiv).#### Integral points on conic log K3 surfaces

*Journal of the European Mathematical Society*, 21 (3), 2019, p. 627–664. (pdf, arXiv).#### On the fibration method for zero-cycles and rational points

## with Olivier Wittenberg,

*Annals of Mathematics*, 183 (1), 2015, p. 229-295 (pdf, arXiv).#### The Hardy-Littlewood conjecture and rational points

## with Alexei Skorobogatov and Olivier Wittenberg,

*Compositio Mathematica*, 150, 2014, p. 2095-2111 (pdf, arXiv).#### Singular curves and the étale Brauer-Manin obstruction for surfaces

## with Alexei Skorobogatov,

*Annales Scientifiques de l'École Normale Supérieure*, 47, 2014, p. 765-778 (pdf, arXiv).#### Homotopy obstructions to rational points

## with Tomer Schlank, In: Alexei Skorobogatov (Ed.), Torsors, Étale Homotopy and Applications to Rational Points, LMS Lecture Notes Series 405, Cambridge University Press, 2013, pp. 280-413 (pdf, arXiv).

#### Deligne's conjecture for unital coherent ∞-operads

## We prove a generalization of Deligne's conjecture for arbitrary unital coherent ∞-operads. With Eduard Balzin,

*in preparation*#### The infinitesimal tangle hypothesis

## We prove an infinitesimal version of the tangle hypothesis, namely, that the cotangent complex of the tangle m-cube monoidal (∞,n)-category is free of rank one, in a suitable sense .

## With Joost Nuiten,

*in preparation*#### Obstruction theory for higher categories

## We develop a Postnikov-based obsrtuction theory for (∞,n)-categories.

## With Joost Nuiten and Matan Prasma,

*submitted*(pdf, arXiv).#### Cartesian Fibrations of (∞,2)-categories

## With Andrea Gagna and Edoardo Lanari,

*submitted*(pdf, arXiv).#### Fibrations and lax limits of (∞,2)-categories

## We develope the notions outer and inner (co)cartesian of (∞,2)-categories and use it to define and study various types of lax and homotopy limits for diagrams taking values in an (∞,2)-category.

## With Andrea Gagna and Edoardo Lanari,

*submitted*(pdf, arXiv).#### Bilimits and bifinal objects

## With Andrea Gagna and Edoardo Lanari,

*Journal of Pure and Applied Algebra*, 226.12, 2022 (pdf, arXiv).#### On the equivalence of all models for (∞,2)-categories

## With Andrea Gagna and Edoardo Lanari,

*Journal of the London Mathematical Society*, 106.3, 2022, p. 1920-1982 (pdf, arXiv).#### Gray tensor products and lax functors of (∞,2)-categories

## We give well-behaved construction of the Gray tensor product in the setting of scaled simpicial sets and show that it is a left Quillen bifunctor, and in particular preserves homotopy colimits in each variable.

## With Andrea Gagna and Edoardo Lanari,

*Advances in Mathematics*, 391, 2021 (pdf, arXiv).#### Ambidexterity and the universality of finite spans

*Proceedings of the London Mathematical society*, 121 (5), 2020, p. 1121–1170. (pdf, arXiv).#### Lax limits of model categories

*Theory and Applications of Categories*, 35, 2020, p. 959–978. (pdf, arXiv).#### The tangent bundle of a model category

## with Joost Nuiten and Matan Prasma,

*Theory and Applications of Categories*, 34, 2019, p. 1039–1072. (pdf, arXiv).#### Quillen cohomology of (∞,2)-categories

## with Joost Nuiten and Matan Prasma,

*Higher Structures*, 3 (1), 2019, p. 17–66, (pdf, arXiv).#### Tangent categories of algebras over operads

## with Joost Nuiten and Matan Prasma,

*The Israel Journal of Mathematics*, 234, 2019, p. 691–742 (pdf, arXiv).#### The abstract cotangent complex and Quillen cohomology of enriched categories

## with Joost Nuiten and Matan Prasma,

*Journal of Topology*, 11 (3), 2018, p. 752-798 (pdf, arXiv).#### Pro-categories in homotopy theory

## with Ilan Barnea and Geoffroy Horel,

*Algebraic and Geometric Topology*, 17 (1), 2017, p. 567-643 (pdf, arXiv).#### An integral model structure and truncation theory for coherent group actions

## with Matan Prasma,

*The Israel Journal of Mathematics*, 221 (2), 2017, p. 511–561 (pdf, arXiv).#### The Grothendieck construction for model categories

## with Matan Prasma,

*Advances in Mathematics*, 281, 2015, p. 1306-1363 (pdf, arXiv).#### Quasi-unital ∞-categories.

*Algebraic and Geometric Topology*, 15 (4), 2015, p. 2303-2381 (pdf, arXiv).

#### The cobordism hypothesis in dimension 1

## We give a proof of the cobordism hypothesis in dimension 1 using the theory of quasi-unital ∞-categories (pdf, arXiv).

#### The section conjecture for graphs and conical curves

## We show that the finite descent obstruction controls the existence of rational points on normal crossing singular curves whose components are all of genus 0, by relating the problem to a fix point property of pro-finite groups acting on pro-finite trees (pdf, arXiv).

#### The universal property of topological Hochschild homology, Chromatic Homotopy, K-Theory and Functors, CIRM Luminy, January 2023.

#### The infinitesimal tangle hypothesis, GdR topologie algébrique, University of Nantes, October 2022.

#### The homotopy theory of quadratic forms, SFB lecture, University of Regensburg, February 2021.

#### New perspectives in hermitian K-theory (three talks), New perspectives on K- and L-theory, University of Muenster, September 2020.

#### Squares represented by a product of three ternary quadratic forms, and a homogeneous variant of a method of Swinnerton-Dyer, Rational points on irrational varieties, Institut Henri Poincaré, June 2019.

#### Supersolvable descent, XXIXèmes Rencontres Arithmétiques de Caen: arithmétique des corps et géométrie arithmétique, Île de Tatihou, May 2019.

#### Ambidexterity and the universality of finite spans, Séminaire de topologie algebrique, Université de Paris 13, April 2018.

#### Zéro-cycles sur les espaces homogènes et problème de Galois inverse, séminaire théorie de nombres de Bordeaux, March 2018.

#### Zéro-cycles sur les espaces homogènes et problème de Galois inverse, séminaire théorie de nombres de Jussieu, March 2018.

#### Ambidexterity and the universality of finite spans, séminaire de topologie, université de Lille, January 2018.

#### Cohomology of higher categories, réunion annuelle du GdR topologie algébrique et applications, October 2017 (pdf).

#### Quillen obstruction theory, séminaire homotopie en géométrie algébrique, October 2017 (pdf).

#### Zero-cycles on homogeneous spaces, Rational points 2017, Franken-Akademie Schloss Schney, July 2017.

#### Quillen cohomology of enriched categories, séminaire de topologie, géométrie et algèbre, Nantes, June 2017 (pdf).

#### Deuxième descente et points rationnels sur les surfaces de Kummer, séminaire arithmétique et géométrie algébrique, université de Strasbourg, January 2017.

#### Second 2-descent and rational points on Kummer surfaces, Rational points and algebraic geometry, Luminy September 2016 (pdf).

#### Higher additivity, higher monoids and the universal property of finite spans., Homotopy Theory Day, Hebrew University of Jerusalem, July 2016 (pdf).

#### Abstract representation theory and the cotangent complex formalism, congrès national de la Société Mathématique de France, Université François Rabelais, Tours, June 2016 (pdf).

#### The cotangent complex formalism, séminaire de physique mathématique et de topologie algébrique, université de Angers, May 2016 (pdf).

#### Pro-categories in homotopy theory, séminaire de Topologie, Université de Lille, April 2016 (pdf).

#### Integral Points on log K3 surfaces, séminaire Variétés Rationnelles, ENS, Paris, April 2016 (pdf).

#### The cobordism hypothesis in dimension 1, séminaire de topologie, géométrie et algèbre, université de Nantes, March 2016.

#### Integral Points on log K3 surfaces, séminaire d'arithmétique et de géométrie algébrique, université Paris-Sud, March 2016 (pdf).

#### Pro-categories in homotopy theory, séminaire de topologie algébrique, université Paris 13, January 2016 (pdf).

#### Rational points on fibered varieties, Interactions between arithmetic and homotopy, Royal Imperial College, London, September 2015 (pdf).

#### The Hasse principle for generalized Kummer varieties, Göttingen-Hannover number theory workshop, Leibniz university of Hannover, July 2015.

#### The descent-fibration method for integral points, Arithmetic geometry, Chow groups and rational points, The Euler Institute, St. Petersburg, June 2015 (pdf).

#### The descent-fibration method for integral points, Heilbronn seminar, university of Bristol, Bristol, May 2015 (pdf).

#### On the fibration method, séminaire Variétés Rationnelles, Institut Henri Poincaré, Paris, February 2015 (pdf).

#### Model fibrations and the Grothendieck correspondence for model categories, Higher Geometric Structures along the Lower Rhine V, Radboud university Nijmegen, June 2014.

#### From linear equations in primes to the fibration method, The intercity number theory seminar, university of Groningen, October 2013.

#### The section conjecture for graphs and applications for singular curves, London-Paris number theory seminar, Royal Imperial College, London, June 2013.

#### The Hasse principle for singular curves with applications for smooth surfaces, séminaire Variétés Rationnelles, Institut Henri Poincaré, Paris, March 2013.

#### Quasi-unital ∞-categories and the cobordism hypothesis in dimension 1, algebraic topology seminar, Radboud university Nijmegen, November 2012 (pdf)

#### Étale homotopy theory, Workshop on Cohomological Methods in Arithmetic Geometry, institut für Mathematik, Zurich, September 2012 (pdf)

#### Étale homotopy and Diophantine equations, Workshop on Arithmetic Geometry and Homotopy Theory, Imperial College, London, May 2012 (pdf)

#### Homotopy obstructions to rational points, joint mathematics meeting, AMS Special Session on Rational Points on Varieties, Boston, January 2012 (pdf)

#### Homotopy obstructions to integral points, Torsors: theory and applications, International Centre for Mathematical Sciences, Edinburgh, January 2011 (pdf)

#### The Étale Homotopy Type and the Local-Global Principle, Rational Points 3, Universität Bayreuth, July 2010 (pdf).

#### Goodwillie approximation for higher categories (after Gijs Heuts), Groupe de travail, Université de Paris 13, 2022 (pdf).

#### Rational points on elliptic surfaces, notes for minicourse given at the IHP, spring 2019 (pdf).

#### Little cube algebras and factorization homology, notes for master course given in Paris 13, spring 2019 (draft under construction, last update 16/04/2019, pdf).

#### Topological Hochschild homology as a cyclotomic spectrum, Groupe de travail sur THH, Université de Paris 6, 2018 (pdf).

#### Factorization homology for topological manifolds, Groupe de travail sur les homologie de factorization, Université de Paris 13, 2017 (pdf).

#### Introduction to stable ∞-categories, Groupe de travail sur les ∞-catégories, Université de Paris 13, 2017 (pdf).

#### (co)Cartesian fibrations, Caesarea 2016 (pdf).

#### Arizona Winter School (2015)

#### Simplicial homotopy theory, Master course, Radboud university 2014 (pdf).

#### Limits, colimits and adjunctions in ∞-categories, Workshop on ∞-Categories, Université Catholique de Louvain, 2013 (pdf).

#### 2-fold complete Segal spaces, Caesarea 2013 (pdf).

#### Étale homotopy and Diophantine equations, Minicourse, Bernoulli centre, EPFL 2012 (link).

#### General relativity, HUJI 2011 (pdf).

#### Basic notions in algebraic topology, TA notes, HUJI 2009-2011 (pdf).

#### ∞-sheaves, Caesarea 2011 (pdf).

#### Elliptic regularity, HUJI 2010 (pdf).

#### Algebraic structures 1, TA notes, HUJI fall 2010 (pdf).

#### Quasi-categories, Caesarea 2010 (pdf).

#### Basic notions in differentials geometry, TA notes, HUJI spring 2009 (pdf).

#### Complex cobordism and formal group laws, Caesarea 2009 (pdf).

#### Classification of framed manifolds, MIT Kan seminar 2009 (pdf).

#### Loop structures on the 3-sphere, MIT babytop seminar 2009 (pdf).

#### Curves, surfaces and the Hasse principle, Summer School on Rational Points, HUJI 2009 (day 1,day 2, day 3).

#### Periodicity in stable homotopy theory 1, HUJI student seminar 2009 (pdf).

#### Periodicity in stable homotopy theory 2, HUJI student seminar 2009 (pdf).

#### Polyhedral groups, HUJI 2009 (pdf).

#### The classification problem for smooth manifolds, HUJI student seminar 2008 (pdf).

#### Galois cohomology and elliptic curves, HUJI 2008 (pdf).

#### Modular curves and modular forms, HUJI 2008 (pdf).